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A normal distribution, also known as a Gaussian distribution, describes how the values of a variable are distributed. It is symmetric, with most values clustering around a central mean, and fewer values appearing as you move away from the mean. This type of distribution graphically forms a bell curve. In our exercise, the random variable X follows a normal distribution with a mean (\(\mu\)) of 50 and a standard deviation (\(\sigma\)) of 7. These values tell us that the average value of X is 50 and the data tends to spread out around this mean by an average of 7 units.

Percentiles are a measure in statistics that indicate the relative standing of a value within a dataset. A specific percentile tells you what percentage of the data falls below a certain value. For instance, if you are looking for the 9th percentile, you are seeking the value below which 9% of your data lies. This is crucial for understanding how values compare within a distribution. Our problem asks us to find the 9th percentile of the normal distribution for X, meaning we need to determine the value below which 9% of X values fall.

The Z-score is a statistical tool that measures how many standard deviations a data point is from the mean. To find the percentile of a normally distributed variable X, we first convert the percentile into a Z-score. This involves finding a Z-score that corresponds to a given cumulative probability using a Z-table. In our problem, we need the Z-score that corresponds to 0.09 (the 9th percentile). From the Z-table, this Z-score is approximately -1.34. Using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] we can solve for X: \[ X = Z \cdot \sigma + \mu \] Plugging in our values: \(-1.34 \cdot 7 + 50 \), we find that X equals approximately 40.62.

Understanding key statistical concepts like normal distribution, percentiles, and Z-scores equips students with essential tools for analyzing data. These concepts are foundational in statistics education because they enable students to interpret data distributions and their respective probabilities accurately. Calculating percentiles, as demonstrated in our exercise, is a practical application that showcases how these principles work together. This knowledge can be applied in various fields, including social sciences, business, and natural sciences, making it a valuable addition to any educational curriculum. By mastering these concepts, students enhance their analytical skills and are better prepared for more advanced statistical studies.